The QCD vacuum probed by overlap fermions

arXiv:hep-lat/0610087v1 13 Oct 2006

DESY 06-177 HU-EP-06/33 LMU-ASC 65/06

The QCD vacuum probed by overlap fermions Volker Weinberg∗ Deutsches Elektronen-Synchrotron DESY, 15738 Zeuthen, Germany Institut für theoretische Physik, Freie Universität Berlin, 14196 Berlin, Germany E-mail: [email protected]

Ernst-Michael Ilgenfritz Institut für Physik, Humboldt Universität zu Berlin, 12489 Berlin, Germany E-mail: [email protected]

Karl Koller Sektion Physik, Universität München, 80333 München, Germany E-mail: [email protected]

Yoshiaki Koma Deutsches Elektronen-Synchrotron DESY, 22603 Hamburg, Germany E-mail: [email protected]

Gerrit Schierholz Deutsches Elektronen-Synchrotron DESY, 22603 Hamburg, Germany John von Neumann-Institut für Computing NIC, 15738 Zeuthen, Germany E-mail: [email protected]

Thomas Streuer Department of Physics and Astronomy, University of Kentucky, Lexington, KY 40506-0055, USA E-mail: [email protected] We summarize different uses of the eigenmodes of the Neuberger overlap operator for the analysis of the QCD vacuum, here applied to quenched configurations simulated by means of the Lüscher-Weisz action. We describe the localization and chiral properties of the lowest modes. The overlap-based topological charge density (with and without UV-filtering) is compared with the results of UV-filtering for the field strength tensor. The latter allows to identify domains of good (anti-)selfduality. All these techniques together lead to a dual picture of the vacuum, unifying the infrared instanton picture with the presence of singular defects co-existent at different scales. XXIVth International Symposium on Lattice Field Theory July 23-28, 2006 Tucson, Arizona, USA ∗ Speaker.

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The QCD vacuum probed by overlap fermions

Volker Weinberg

1. Introduction The understanding of the vacuum structure of QCD is an important part of theoretical particle physics. The distribution of the topological charge density is believed to describe nonperturbative phenomena like the large η ′ mass through the axial U(1) anomaly and the spontaneous breaking of chiral symmetry. The relation to confinement is less obvious. The instanton picture models the QCD vacuum as a random (quenched) or interacting (full QCD) instanton-antiinstanton system, whereas the understanding of confinement seems to require singular fields emanating from lowerdimensional defects [1]. Recent advances in implementing chiral symmetry on the lattice have made it possible to find, from first principles, manifestations of these dual aspects of vacuum structure. Overlap fermions are described by the Neuberger-Dirac operator DN constructed as a solution of the Ginsparg-Wilson relation. They are the cleanest known implementation of lattice fermions. For gauge fields free of dislocations, the index theorem is unambiguously realized and the spectrum consists of chiral zero modes and non-chiral non-zero modes. While the number and chirality of zero modes provides a value for the total topological charge, the non-chiral non-zero modes appear in pairs of eigenvalues ±iλ . For our methodical study we use quenched configurations simulated by means of the LüscherWeisz action. In the following table we list the statistics of lattices used in our investigation. We show also the topological susceptibility χt . To set the scale, we use the quenched pion decay constant fπ = 91(2) MeV.

β 8.45 8.45 8.45 8.10 8.00

a [fm]

lattice size

L [fm]

0.105(2) 0.105(2) 0.105(2) 0.138(3) 0.154(4)

123 × 24

1.26 1.68 2.52 1.66 2.46

163 × 32 243 × 48 123 × 24 163 × 32

V [fm4 ]

χt [fm−4 ]

confs.

modes

5.04 15.93 80.66 15.04 73.72

0.517(38) 0.533(38) 0.522(48) 0.647(58) 0.607(21)

437 400 250 251 2156

50 136-151 144-177 137-154 160-178

After presenting results concerning localization and chiral properties of individual eigenmodes in different parts of the spectrum up to λ = 400 MeV, we describe two specific applications of the overlap Dirac operator. One is aiming to extract a UV-filtered field strength tensor [2] in order to assess the local degree of (anti-)selfduality. The other uses the fermionic definition of topological charge density [3]. We interpret the respective structures and their dimensionality.

2. Localization, dimensionality and local chirality of the lowest eigenmodes Recently, the localization, the eventually lower dimensionality and the local chirality of the lowest eigenmodes of the Dirac operator have attracted a lot of interest [4, 5]. The reason for this activity is that these modes might be pinned down by some topological defects. This is inspired by certain confinement mechanisms, e.g. the center vortex mechanism. In model field configurations of thick vortices the zero modes have been seen [6] localized along the vortices with additional peaks at the intersections. The particular interest in the lowest eigenmodes is motivated by the 2


Volker Weinberg

observation that the lightest hadron propagators are very well approximated by taking only O(50) lowest eigenmodes in the quark propagator into account. A useful measure to quantify the localization of eigenmodes is the inverse participation ratio (IPR), I = V ∑x ρ (x)2 , with the scalar density ρ (x) = ψλ† (x)ψλ (x) for eigenfunctions with eigenvalues iλ , which are conventionally normalized as ∑x ρ (x) = 1. In Fig. 1 (a) we plot the IPR averaged over bins with a bin width ∆λ = 50 MeV and for the zero modes considered separately. The average IPR shows an L and a dependence for zero modes and for non-zero modes only in the range λ < 150 MeV. Beyond that the average IPR corresponds to 4-dimensionally extended states. In the theory of the metal-insulator transition, a more detailed description of the localization properties at different “heights” of the wave functions has led to consider generalized IPRs like Ip = ∑x ρ (x) p , averaged over parts of the spectrum and different realizations of disorder. In any physical application, dimensionalities d ∗ (p) different from the embedding dimension d and eventually varying with p, can be inferred from the volume scaling, I p ∝ L−d(p−1) ∗ I p ∝ L−d (p)(p−1) I p ∝ const

metallic phase, critical region, insulator phase,

extended wave functions, multifractal wave functions, localized wave functions.

While the dimension for the metallic or insulator phase is d or 0, respectively, the critical multifractal region in between is characterized by a multitude of fractal dimensions d ∗ (p) and also by critical level statistics [7]. 4

14

123 × 24, β = 8.10

12

163 × 32, β = 8.00

3

10

163 × 32, β = 8.45

2.5

3.5

123 × 24, β = 8.45

243 × 48, β = 8.45

d∗ (p)

IPR

16

8

2

6

1.5

4

1

2

0.5

0

p=2 p=3 p=4 p=10 p=20 p=30

0

0

50

100

150 200 λ [MeV]

250

300

350

0

50

100

150

200

250

300

350

400

λ [MeV]

(a)

(b)

Figure 1: (a) The average IPR is shown for zero modes and for non-zero modes in λ bins of width 50 MeV for the five ensembles. (b) The fractal dimensionality d ∗ (p) obtained from fits of the volume dependence of the averages of the generalized I p is presented for zero modes and for non-zero modes in λ bins of width 50 MeV for the three ensembles with different volumes and common β = 8.45.

From a fit of the volume dependence of Ip we find that all eigenstates with λ < 200 MeV in this sense belong to a critical regime. From the standard IPR, I = V I2 , we conclude that the dimensionality of the zero modes is close to two. In the next bins of width 50 MeV each, the dimensionality is near three, in agreement with [4] and disagreement with [5] 1 . The IPR of the higher modes is independent of V and a, describing the observation that the modes freely extend throughout d ∗ (2) . d = 4 dimensions. 1 However, in [4] the Asqtad Dirac operator on quenched

Symanzik improved gauge fields is used, while the analysis in [5] is done with overlap fermions on quenched SU(2) configurations generated with the standard Wilson action.

3


Volker Weinberg

Looking at the higher moments of IPR one can see a decrease of the multifractal dimension with increasing p as shown in Fig. 1 (b). This reflects the fact that I p with p > 2 explores regions of higher scalar density. For p = 20 . . . 30 the emerging fractal dimension seems to converge. The envelope reveals that the maxima of the scalar density of zero modes and the lowest non-zero modes up to λ = 100 MeV are characterized by d ∗ (p) < 1 localization (pointlike peaks), whereas in the spectral region 100 MeV < λ < 400 MeV, d ∗ (p) = 2 or 3 characterizes the maxima of the scalar density. In contrast to this, the standard IPR alone would describe the modes in the spectral region λ > 150 MeV as four-dimensional. For the zero modes, the results favor an interpretation in the vortex picture above, but that needs direct confirmation.

(a)

(b)

(c)

(d)

Figure 2: Normalized histograms of the local chirality X(x) in the Q = 0 subsample (consisting of 37 configurations) among the 163 × 32 lattices generated at β = 8.45. (a) and (c): for the lowest ten pairs (with positive and negative λ ) averaged over the subsample. (b) and (d): for all modes from the subsample, averaged over eight λ bins of width 100 MeV. (a) and (b): with a cut for 1/100 of the sites with biggest scalar density ρ (x). (c) and (d): with a cut for 1/2 of the sites.

The local chiral density ρλ 5 (x) = ψλ† (x)γ5 ψλ (x) with ∑x ρλ 5 (x) = ±1 or 0 for chiral or nonchiral modes, respectively, is equally important. Whereas the zero modes are entirely chiral, the non-zero modes with globally vanishing chirality may still have a rich local chirality structure correlated with the local (anti-)selfduality of the background gauge field. A measure proposed by Horvath [8] to describe the local chirality ρλ ± (x) = ψλ† (x)P± ψλ (x) (with projectors P± = (1 ± γ5 ) /2 onto positive/negative chirality) is given by X (x) =

4 arctan (rλ (x)) − 1 ∈ [−1, +1] , π

(2.1)

with rλ (x) = ρλ + (x)/ρλ − (x). For the chiral zero modes X (x) ≡ ±1, while for the non-chiral nonzero modes X (x) is a strongly varying field. In Fig. 2 normalized histograms with respect to X (x) are shown for the Q = 0 subsample (37 configurations) of the 163 × 32 lattices generated at β = 8.45, with cuts including 1/100 and 1/2 of the sites according to the biggest ρ (x). For zero modes, the histograms would be concentrated at X = ±1 irrespective of any cut. For the non-zero modes, it turns out that with a more restrictive cut in ρ (x) the local chirality, i.e. the concentration of the histogram over the included lattice points, can be driven towards X = ±1. A similar enhancement can be seen towards the lower modes. For the lowest two pairs of non-zero modes all sites with ρ (x) above median show the two chiral peaks. 4


Volker Weinberg

3. The UV-filtered gluonic field strength tensor and the topological charge density The lowest eigenmodes of the overlap operator can directly be used as a filter of the underlying gauge fields. Following a method proposed by Gattringer [2], the gluonic field strength tensor can be represented as a a Fµν (x) ∝ ∑ λ j2 f µν (x) j . (3.1) j

a (x) = − i ψ † (x)γ γ T a ψ (x) to the field strength tensor The contribution of the j-th eigenmode f µν j µ ν j 2 j a (x) is projected out by means of an appropriate combination of γ -matrices with the SU(3) Fµν generator T a . To study the local (anti-)selfduality of the filtered field strength tensor, we analyze the ratio r(x) = (s(x) ˜ − q(x))/( ˜ s(x) ˜ + q(x)) ˜ , (3.2)

˜ = Tr Fµν (x) F˜µν (x) with the action density s(x) ˜ = Tr Fµν (x) Fµν (x) and the topological density q(x) of the filtered field strength summing in Eq. (3.1) over different sets of low-lying non-zero modes. Using the ratio r(x) we do not need to know the normalization factor of s(x) ˜ and q(x). ˜ Mapping r(x) to R(x) ∈ [−1, +1] analogously to Eq. (2.1), one would get R(x) ≡ ±1 for totally (anti-)selfdual fields. In Fig. 3 we present histograms with respect to R(x) applying two different cuts with respect to the filtered action density s(x). ˜ We observe peaks at R = ±1 that become weaker with the inclusion of more modes. The peaks become more pronounced again when one concentrates on lattice points above some cut with respect to the filtered action density s(x). ˜

0.25

0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

0.2 0.15 0.1 0.05 0.5 00

2

0 4

6 8 10 12 14 16 18 20 Nr. of modes

-0.5

R

(a)

0.5 1 5 10 20 30 40 50 60 % of lattice points 70 80 90 100

0 -0.5

R

(b)

Figure 3: Normalized histograms with respect to the local (anti-)selfduality R(x) of the field strength tensor in the Q = 0 subsample of 37 configurations generated on the 163 × 32 lattice at β = 8.45, (a) taken over the full volume, but depending on the number of non-zero modes (2 - 20) included in the filter, (b) applying various cuts according to the action density for a fixed number (20) of eigenmodes included in the filter.

Alternatively to q(x) ˜ based on the filtered field strength tensor we also examine the topologi cal charge density as given by the formula q(x) = − Tr γ5 1 − 2a DN (x, x) [3] which holds for any γ5 -Hermitean Dirac operator satisfying the Ginsparg-Wilson relation. We use it in a twofold way [9, 10]. (i) We directly calculate the trace of the overlap operator. The “all-scale” density q(x) computed in this numerically very expensive way includes charge fluctuations at 5


Volker Weinberg

all scales down to the lattice spacing a. (ii) We use the mode-truncated spectral representation qλcut (x) = − ∑|λ |